109 citations · 567 across the 25 of their papers we have counts for
29 papers
Spherical Cap Discrepancy -- Blessing of Dimensionality and a Balanced Large-Cap Variant
Johann S. Brauchart, Josef Dick, Friedrich Pillichshammer
We prove that the information complexity (i.e., the inverse) of the classical spherical cap discrepancy on the -dimensional sphere decreases with dimension…
On the lower bounds for the spherical cap discrepancy
Dmitriy Bilyk, Johann S. Brauchart
We start by providing a very simple and elementary new proof of the classical bound due to J. Beck which states that the spherical cap -discrepancy of any points…
Hyperuniform point sets on projective spaces
Johann S. Brauchart, Peter J. Grabner
We extend the notion of hyperuniformity to the projective spaces , , , and . We show that hyperuniformity im…
Needlets Liberated
Johann S. Brauchart, Peter J. Grabner, Ian H. Sloan +1
Spherical needlets were introduced by Narcowich, Petrushev, and Ward to provide a multiresolution sequence of polynomial approximations to functions on the sphere. The needlet cons…
Complete minimal logarithmic energy asymptotics for points in a compact interval: a consequence of the discriminant of Jacobi polynomials
Johann S. Brauchart
The electrostatic interpretation of zeros of Jacobi polynomials, due to Stieltjes and Schur, enables us to obtain the complete asymptotic expansion as of the minimal…
Weighted -Norms of Gegenbauer polynomials
Johann S. Brauchart, Peter J. Grabner
We study integrals of the form \begin{equation*} \int_{-1}^1(C_n^{(λ)}(x))^2(1-x)^α(1+x)^β\, dx, \end{equation*} where denotes the Gegenbauer-polynomial of index …
Hyperuniform point sets on the sphere: probabilistic aspects
Johann S. Brauchart, Peter J. Grabner, Wöden B. Kusner +1
The concept of hyperuniformity has been introduced by Torquato and Stillinger in 2003 as a notion to detect structural behaviour intermediate between amorphous disorder and crystal…
Hyperuniform point sets on the sphere: deterministic constructions
Johann Brauchart, Peter Grabner, Wöden Kusner
We study a generalisation of the concept of hyperuniformity to spheres of arbitrary dimension. It is shown that QMC-designs (and especially spherical designs) are hyperuniform in o…
Logarithmic and Riesz Equilibrium for Multiple Sources on the Sphere --- the Exceptional Case
Johann S. Brauchart, Peter D. Dragnev, Edward B. Saff +1
We consider the minimal discrete and continuous energy problems on the unit sphere in the Euclidean space in the presence of an external field due…
Random Point Sets on the Sphere --- Hole Radii, Covering, and Separation
Johann S. Brauchart, Alexander B. Reznikov, Edward B. Saff +3
Geometric properties of random points distributed independently and uniformly on the unit sphere with respect to surface area measure ar…
Spatial low-discrepancy sequences, spherical cone discrepancy, and applications in financial modeling
Johann S. Brauchart, Josef Dick, Lou Fang
In this paper we introduce a reproducing kernel Hilbert space defined on as the tensor product of a reproducing kernel defined on the unit sphere $\mathbb{S}^{d}…
Covering of spheres by spherical caps and worst-case error for equal weight cubature in Sobolev spaces
Johann S. Brauchart, Josef Dick, Edward B. Saff +3
We prove that the covering radius of an -point subset of the unit sphere is bounded above by a power of the worst-case error for equal weight cubatur…
Distributing many points on spheres: minimal energy and designs
Johann S. Brauchart, Peter J. Grabner
This survey discusses recent developments in the context of spherical designs and minimal energy point configurations on spheres. The recent solution of the long standing problem o…
An Electrostatics Problem on the Sphere Arising from a Nearby Point Charge
Johann S. Brauchart, Peter D. Dragnev, Edward B. Saff
For a positively charged insulated d-dimensional sphere we investigate how the distribution of this charge is affected by proximity to a nearby positive or negative point charge wh…
Riesz external field problems on the hypersphere and optimal point separation
Johann S. Brauchart, Peter D. Dragnev, Edward B. Saff
We consider the minimal energy problem on the unit sphere in the Euclidean space in the presence of an external field , where the energy arises…
"Magic" numbers in Smale's 7th problem
Rachele Nerattini, Johann S. Brauchart, Michael K. -H. Kiessling
Smale's 7-th problem concerns N-point configurations on the 2-dim sphere which minimize the logarithmic pair-energy V_0(r) = -ln r averaged over the pairs in a configuration; here,…
QMC designs: optimal order Quasi Monte Carlo Integration schemes on the sphere
Johann S. Brauchart, Edward B. Saff, Ian H. Sloan +1
We study equal weight numerical integration, or Quasi Monte Carlo (QMC) rules, for functions in a Sobolev space with smoothness parameter defined over the unit s…
A characterization of Sobolev spaces on the sphere and an extension of Stolarsky's invariance principle to arbitrary smoothness
J. Brauchart, J. Dick
In this paper we study reproducing kernel Hilbert spaces of arbitrary smoothness on the sphere . The reproducing kernel is given by an integr…
The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere
J. S. Brauchart, D. P. Hardin, E. B. Saff
We survey known results and present estimates and conjectures for the next-order term in the asymptotics of the optimal logarithmic energy and Riesz -energy of points on the…
Point sets on the sphere with small spherical cap discrepancy
Christoph Aistleitner, Johann Brauchart, Josef Dick
In this paper we study the geometric discrepancy of explicit constructions of uniformly distributed points on the two-dimensional unit sphere. We show that the spherical cap discre…
Explicit formulas for the Riesz energy of the th roots of unity
J. S. Brauchart
The paper Brauchart, Hardin and Saff [Bull. Lond. Math. Soc. 41(4) (2009)] gives the complete asymptotic expansions of the Riesz -energy of the th roots of unity which form a…
Optimal Discrete Riesz Energy and Discrepancy
J. S. Brauchart
The Riesz -energy of an -point configuration in the Euclidean space is defined as the sum of reciprocal -powers of all mutual distances in this system. In…
A Fascinating Polynomial Sequence arising from an Electrostatics Problem on the Sphere
J. S. Brauchart, P. D. Dragnev, E. B. Saff +1
A positive unit point charge approaching from infinity a perfectly spherical isolated conductor carrying a total charge of +1 will eventually cause a negatively charged spherical c…
A simple Proof of Stolarsky's Invariance Principle
Johann S. Brauchart, Josef Dick
Stolarsky [Proc. Amer. Math. Soc. 41 (1973), 575--582] showed a beautiful relation that balances the sums of distances of points on the unit sphere and their spherical cap $\mathbb…